In this paper, we derive and study a new class of thermodynamically consistent Navier–Stokes–Cahn–Hilliard systems with dynamic boundary conditions. Our model derivation is based on local mass balance laws in the bulk and on the surface in which the mass fluxes are still to be determined. Arguing similarly as in [1]}, after considering local energy dissipation laws, we apply the Lagrange multiplier approach to complete our model derivation by identifying the unknown mass fluxes on the macroscopic level.
The resulting Navier–Stokes–Cahn–Hilliard (NSCH) model reads as follows:
\(&\partial _{t}\big (\rho (\phi ) {v}\big )+ \textnormal {div}\big ( {v}\otimes (\rho (\phi ){v}+ J) \big )- \textnormal {div}\big ( 2\nu (\phi ) \, \big )+ \nabla p= - \varepsilon \, \textnormal {div}\big (\nabla \phi \otimes \nabla \phi \big )&&\;\;\text{in $Q$},\\[1ex]&\textnormal {div}\, {v}= 0,&&\;\;\text{in $Q$},\\[1ex]&\rho (\phi )= \tilde{\rho }_1 \frac{1+\phi }{2}+ \tilde{\rho }_2 \frac{1-\phi }{2},\quad J= - \tfrac{1}{2} (\tilde{\rho }_2 - \tilde{\rho }_1)m_\Omega (\phi ) \nabla \mu &&\;\;\text{in $Q$},\\[1ex]&\partial _{t}\phi + \textnormal {div}(\phi {v})= \textnormal {div}\big ( m_\Omega (\phi ) \nabla \mu \big )&&\;\;\text{in $Q$},\\[1ex]&\mu = -\varepsilon \Delta \phi + \varepsilon ^{-1} F^{\prime }(\phi )&&\;\;\text{in $Q$},\\[1ex]&\partial _{t}\psi + \textnormal {div}_\Gamma (\psi {v}_\tau )= \textnormal {div}_\Gamma \big ( m_\Gamma (\psi ) \nabla _\Gamma \theta \big ) - \beta m_\Omega (\psi )\, \partial _\mathbf {n}\mu &&\;\;\text{on $\Sigma $},\\[1ex]&\theta = - \delta \kappa \Delta _\Gamma \psi + \delta ^{-1} G^{\prime }(\psi )+ \varepsilon \partial _\mathbf {n}\phi &&\;\;\text{on $\Sigma $},\\[1ex]&{v}\cdot \mathbf {n}= 0,\qquad \phi \vert _\Gamma = \psi ,\qquad {\left\lbrace \begin{array}{ll}Lm_\Omega (\psi ) \, \partial _\mathbf {n}\mu = \beta \theta - \mu &\text{if $L\in [0,\infty )$},\\m_\Omega (\psi )\, \partial _\mathbf {n}\mu = 0&\text{if $L = \infty $}\end{array}\right.}&&\;\;\text{on $\Sigma $},\\[1ex]& \big [2\nu (\psi ) (\cdot \mathbf {n}) + \gamma (\psi ) {v}\big ]_\tau = \big [ -\psi \nabla _\Gamma \theta + \tfrac{1}{2} \, (J\cdot \mathbf {n}) \, {v}\big ]_\tau &&\;\;\text{on $\Sigma $},\\[1ex]&{v}\vert _{t=0} = {v}_0,\quad \phi \vert _{t=0} = \phi _0&&\;\;\text{in $\Omega $},\\[1ex]&\psi \vert _{t=0} = \psi _0&&\;\;\text{on $\Gamma $},\) 
